Weierstrass型函数关联分形集的测度和维数的研究

This project intends to study the measure and dimension of fractal sets (including image sets, graphs, level sets and inverse image sets) associated with the Weierstrass-type function, which is obtained by replacing cosine function in the Weierstrass function with other periodic function. The Weierstrass-type function is continuous nowhere differentiable, which is an important research object of fractal geometry. The fractal properties of the Weierstrass-type function are very important in both theory and application. The project will take the Moran sets in Euclidean space and the symbolic space as models, and carry out the research by combining techniques in fractals and in Combinatorial mathematics. The main consideration is the replaced function being the Rademacher function and the Takagi function(sawtooth function). The key point is to present the dimension formula of Moran sets (with overlaps or with the infimum of compression ratios being zero). These questions involve the core of the research of fractal geometry. The related research results will be helpful to understand deeply the Weierstrass-type function, and then promote research and development of fractal functions.

本项目拟研究Weierstrass型函数(将Weierstrass函数中的余弦函数替换为其他周期函数而得到)所关联分形集(包括象集,图像,水平集和逆象集)的测度和维数.Weierstrass型函数是处处连续处处不可微,也是分形几何中一类重要研究对象,其分形性质在理论和应用上都极具重要性.本项目拟以欧氏空间和符号空间上的Moran集为模型,将分形和组合数学中的技巧相结合来展开研究,主要考虑被替代函数为Rademacher函数和Takagi函数(锯齿函数)两种情形.项目关键点是给出相应情形(具有重叠结构或压缩比下确界为0)Moran集的维数公式.这些问题涉及分形几何研究的核心内容,相关研究结果有助于深入认识Weierstrass型函数,从而促进分形函数的研究和发展.

Weierstrass函数图像的分形维数是分形几何的一个热点问题,它的研究进展一直为人们所关注.本项目研究Weierstrass型函数$f(x)=\suma_kg(b_kx)$图像及相关性质,这里${a_k}$为$\mathbb{R}^d$中的向量序列.当$g$为向量值Rademacher函数时,我们考虑其象集值域,得到了其象集的分形维数;我们也研究下面两种集合的拓扑性质:(1)$g$为实直线上的标准Rademacher函数时,$f(x)$的发散水平集,(2)$g$为锯齿函数时,$f(x)$局部水平集所诱导出的重分形集,证明了它们为residual集的充分且必要条件.本项目的进展有助于人们深入理解维数猜想,从而促进分形函数的研究和发展。